I've been watching Season 1 of the original Hawaii Five-0, and when McGarrett takes his jacket off, he's wearing this weird shoulder holster.
How does it stay on? I've been all around the internet and can't find a good answer. My best guess is whatever he uses to keep his hair up like that is also keeping his shoulder holster up.

Also, McGarrett is a serious coffee fiend in the show -- I don't think I picked up on that as a kid.

There are so many different types of shoulder holsters, most seem to be horizontal ones but the originals were vertical ones. It appears Jack Lord wore a vertical revolver shoulder holster. In this photo the strap over his left and right shoulder are visible.
https://www.pinterest.com/pin/441845413438726234/

If he was carrying a stage/prop pistol made of rubber or plastic there was next to no weight and if he wasn't moving very much it would probably have sat on his shoulder. Or an adhesive to stick it to his shirt. BKinsey worked in the industry, he might know.

I can't see the link you posted. Sounds like a more traditional shoulder holster. The one he's wearing in the early episodes is clearly just on one side. But it's McGarrett, so anything possible. in the show, he appears to have mastered long range snub nose marksmanship. Which is also awesome.

Apparently the Smithsonian has the shoulder holster and the Colt Cobra Jack Lord wore. Looks like it just sat on his left shoulder. Your eyes didn't deceive you. It's TV so of course it's not real.
https://americanhistory.si.edu/collecti ... mah_680725

Cool find!